Question

Show that `|(z - 2)/(z - 3)|` = 2 represents a circle. Find its centre and radius.

Answer 1

Given that: `|(z - 2)/(z - 3)|` = 2 

Let z = x + iy

∴ `|(x + iy - 2)/(x + iy - 3)|` = 2

⇒ `|((x - 2) + iy)/((x - 3) + iy)|` = 2  ......`[because |a/b| = |a|/|b|]`

⇒ `|(x - 2) + iy| = 2|(x - 3) + iy|`

⇒ `sqrt((x - 2)^2 + y^2) = 2sqrt((x - 3)^2 + y^2)`

Squaring both sides, we get

(x – 2)² + y² = 4[(x – 3)² + y²]

⇒ x² + 4 – 4x + y² = 4[x² + 9 – 6x + y²]

⇒ x² + y² – 4x + 4 = 4x² + 4y² – 24x + 36

⇒ 3x² + 3y² – 20x + 32 = 0

⇒ `x^2 + y^2 - 20/3 x + 32/3` = 0

Here g = `(-10)/3`, f = 0

r = `sqrt(g^2 + f^2 - "c")`

= `sqrt(100/9 + 0 - 32/3)`

= `sqrt(100/9 - 32/3)`

= `sqrt((100 - 96)/9)`

= `sqrt(4/9)`

= `2/3`

Hence, the required equation of the circle is `x^2 + y^2 - 20/3 x + 32/3` = 0

Centre = (–g, –f) = `(10/3, 0)` and r = `2/3`.